1,3

Number of necklaces of 2 colors with 2n beads and n-1 black ones. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 03 2002

Number of rooted planar binary trees up to reflection (trees with n internal nodes, or a total of 2n+1 nodes). - Antti Karttunen, Aug 19 2002

Number of even permutations avoiding 132.

Number of Dyck paths of length 2n having an even number of peaks at even height. Example: a(3)=3 because we have UDUDUD, U(UD)(UD)D, and UUUDDD, where U=(1,1), D=(1,-1), and the peaks at even height are shown between parentheses. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 13 2004

Number of planar trees (A002995) on n edges with one distinguished edge. - David Callan (callan(AT)stat.wisc.edu), Oct 08 2005

P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183.

T. D. Noe, Table of n, a(n) for n=1..200

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

T. Mansour, Counting occurrences of 132 in an even permutation.

G.f.: (2-2*x-sqrt(1-4*x)-sqrt(1-4*x^2))/x/4. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 26 2003

A007595 := n -> (1/2)*(Cat(n) + (`mod`(n, 2)*Cat((n-1)/2))); Cat := n -> binomial(2*n, n)/(n+1);

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-1)/# ] &)/@Intersection[Divisors[2n], Divisors[n-1]])/(2n), {n, 2, 32}] or Table[If[EvenQ[n], cat[n]/2, (cat[n] +cat[(n-1)/2])/2], {n, 24}] with cat[n]=A000108

a(n)=A047996(2*n, n-1) for n>= 1, and a(n)=A072506(n, n-1) for n>=2. Occurs in A073201 as the rows 0, 2, 4, etc. (with a(0)=1 included). Cf. also A003444, A007123.

Cf. A000150.

Adjacent sequences: A007592 A007593 A007594 this_sequence A007596 A007597 A007598

Sequence in context: A079120 A092566 A036719 this_sequence A035353 A075214 A070766

nonn,easy

njas

Description corrected by Reiner Martin and Wouter Meeussen, Aug 04 2002.